Gap-SBM: A new conceptualization of the shifted boundary method with optimal convergence for the Neumann and Dirichlet problems

We propose and mathematically analyze a new Shifted Boundary Method for the treatment of Dirichlet and Neumann boundary conditions, with provable optimal accuracy in the L²- and H¹-norms of the error. The proposed method is built on three stages. First, the distance map between the SBM surrogate boundary and the true boundary is used to construct an approximation to the geometry of the gap between the two. Then, the representations of the numerical solution and test functions are extended from the surrogate domain to the such gap. Finally, approximate quadrature formulas and specific shift operators are applied to integrate a variational formulation that also involves the fields extended in the gap. An extensive set of two- and three-dimensional tests demonstrates the theoretical findings and the overall optimal performance of the proposed method.

Duke Scholars

Author Guglielmo Scovazzi Civil and Environmental Engineering